42 Triangle Congruenceby SSS and SAS

Mathematics Florida Standards

MAFS.912.G-SRT.2.5 Use congruence... criteria fortriangles to solve problems and prove relationships Ingeometric figures.

MP1,MP 3, MP4,MP7

JObjective To prove two triangles congruent using the SSS and SAS Postulates

SfGetting Ready!

Are the triangles below congruent? How do you know?

<>' X C

How con you tellwhether these

triangles arecongruent? In thislesson, you will learnthe least amount

of informationrequired to tell iftwo triangles arecongruent.

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MATHEMATICAL

PRACTICESIn the Solve It, you looked for relationships between corresponding sides and

angles. In Lesson 4-1, you learned that if two triangles have three pairs of congruentcorresponding angles and three pairs of congruent corresponding sides, then the

triangles are congruent.

If you know...

/-F= u fg = 7k

^H= /LL FH = 1L

... then you know AFGH = AJKL.

However, this is more information about the corresponding parts than you need toprove triangles congruent.

Essential Understanding You can prove that two triangles are congruentwithout having to show that all corresponding parts are congruent. In this lesson,

you will prove triangles congruent by using (1) three pairs of corresponding sidesand (2) two pairs of corresponding sides and one pair of corresponding angles.

226 Chapter 4 Congruent Triangles

F".You have two pairsof congruent sides.What else do youneed?

You need a third pair ofcongruent correspondingsides. Notice that

the triangles share a

common side, W.

Postulate 4-1 Side-Side-Side (SSS) Postulate

Postulate

If the three sides of one

triangle are congruent to

the three sides of another

triangle, then the two

triangles are congruent.

If...

AB = W,^ = ̂,AC=^B E

Then ...

AABC s ADEF

As described in Chapter 1, a postulate is an accepted statement of fact. The Side-Side-Side Postulate is perhaps the most logical fact about triangles. It agrees with the notionthat triangles are rigid figures; their shape does not change until pressure on their sides

forces them to break. This rigidity property is important to architects and engineerswhen they build things such as bicycle frames and steel bridges.

Using SSS

Given: LM = NP, IP = NM

Prove: ALMN - ANPL

LM = NP

Given

LN^LN

Reflexive Prop, of

LP = NM

Given

ALMN = ANPL

SSS

Gof If? 1. Given: BC = BF, CD = FD

Prove: ABCD = ABFD

D

C PowerGeometiy.com Lesson 4-2 Triangle Congruence by SSS and SAS 227

You can also show relationships

between a pair of correspondingsides and an included angle.

Ihe word included refers to the

angles and the sides of a triangle

as shown at the right.

LA is induded

between BA

and AC.

BC is included

between LB

and lC.

Postulate 4-2 Side-Angle-Side (SAS) Postulate

Postulate

If two sides and the

included angle of one

triangle are congruent to

two sides and the included

angle of another triangle,

then the two triangles arecongruent.

If...

AB ='^, LA = LD,

AC = W

Then ...

AABC = ADEF

You likely have used the properties of the Side-Angle-Side Postulate before. Forexample, SAS can help you determine whether a box will fit through a doorway.

Suppose you keep your arms at a fixed angle as you move from the box to the doorway.

Ihe triangle you form with the box is congruent to the triangle you form with thedoorway. Ihe two triangles are congruent because two sides and the included angle ofone triangle are congruent to the two sides and the included angle of the other triangle.

228 Chapter 4 Congruent Triangles

Do you need anotherpair of congruentsides?

Look at the diagram.

The triangles share Of.So, you already have twopairs of congruent sides.

Problem 2

What should youlook for first, sides or

angles?Start with sides. If youhave three pairs ofcongruent sides, use SSS.If you have two pairs ofcongruent sides, lookfor a pair of congruentincluded angles.

I Using SASWhat other information do you need to prove

ADEF = AFGD by SAS? Explain.

The diagram shows that EF = GD. Also, DF = DF bythe Reflexive Property of Congruence. To prove that

ADEF = AFGD by SAS, you must have congruent

included angles. You need to know that AEFD = AGDF.

Got It? 2. What other information do you need to proveALEB = ABNLbySAS?

Recall that, in Lesson 1-6, you learned to construct

segments using a compass open to a fixed angle. Nowyou can show that it works. Similar to the situation with

the box and the doorway, the Side-Angle-Side Postulatetells you that the triangles outlined at the right are

congruent. So, AB = CD.

Identifying Congruent Triangles

Would you use SSS or SAS to prove the triangles congruent? If there is not enoughinformation to prove the triangles congruent by SSS or SAS, write not enough

information. Explain your answer.

0 ----A □

Use SAS because two pairs ofcorresponding sides and their includedangles are congruent.

There is not enough information; two pairs ofcorresponding sides are congruent, but oneof the angles is not the included angle.

8 Q

Use SSS because three pairs ofcorresponding sides are congruent.

Use SSS or SAS because all three pairs ofcorresponding sides and a pair of includedangles (the vertical angles) are congruent.

Got It? 3. Would you use SSS or SAS to prove the triangles at theright congruent? Explain.

PowerGeometry.com Lesson 4-2 TViangle Congruence by SSS and SAS 229

Lesson Check

Do you know HOW?1. In APEN, name the angle that is included between

the given sides,

a. PE and EN b. NP and PE

2. In AHAT, between which sides is the given angleincluded?

a. AH b. AT

Name the postulate you would use to prove the

triangles congruent.

Do you UNDERSTAND?

5. Compare and Contrast How are the SSS Postulate

and the SAS Postulate alike? How are they different?

6. Error Analysis Your friend thinks that the trianglesshown below are congruent by SAS. Is your friend

correct? Explain.

7. Reasoning A carpenter trims a triangular peakof a house with three 7-ft pieces of molding. Ihe

carpenter uses 21 ft of molding to trim a second

triangular peak. Are the two triangles formed

congruent? Explain.

Practice and Problem-Solving Exercises

Practice @ 8. Developing Proof Copy and complete theflow proof.

Given: Jk = LM, Jm = LK -/

Prove: AJKM = ALMK

MATHEMATICAL

PRACTICES

M

JKsLM

Given

JM = LK

a. ?

^ See Problem 1.

K

KM^KM

b. ?

c._?_ = d.

SSS

^ Given: IE = GH, EE s HF,Proof p is midpoint of GI

Prove: AEFl = AHFG

H

10. Given: WZ = ZS = '^ = dW

Pf5?f Prove: AWZD = ASDZ

W

230 Chapter 4 Congruent Triangles

What other information, if any, do you need to prove the two trianglescongruent by SAS? Explain.

12.

W

L

11.

R

^ See Problem 2.

M

W

Would you use SSS or SAS to prove the triangles congruent? If there is notenough information to prove the triangles congruent by SSS or SAS, write not

enough information. Explain your answer.

^ See Problem 3.

13. 14.

Apply @ 15. Think About a Plan You and a friend are cutting triangles out of felt for an artproject. You want all the triangles to be congruent. Your friend tells you that eachtriangle should have two 5-in. sides and a 40° angle. If you follow this rule, will allyour felt triangles be congruent? Explain.

• How can you use diagrams to help you?

• Which postulate, SSS or SAS, are you likely to apply to the given situation?

1^. Given: BC = DA, Z.CBD = /LADBProof ^

17. Given: Xis the midpoint of AG and Ni?.Proof I

Prove: AAWX= LGBX

A

Use the Distance Formula to determine whether A ABC and ADEF are

congruent. Justify your answer.

18. A(l,4), B{5,5), C(2, 2);

D(-5, I),£(-1,0),H-4. 3)

19. AC3, 8), B(8,12), C(10, 5); 20. A(2, 9), 5(2, 4), C(5, 4);

D(3, -1), £(7, -7), H12, -2) D(l, -3), £(1,2), F{-2,2)

21. Writing List three real-life uses of congruent triangles. For each real-life use,

describe why you think congruence is necessary.

C PowerGeometry.corn { Lesson 4-2 Triangle Congruence by SSS and SAS 231

22. Sierpinski's Triangle Sierpinski's triangle is a famous

geometric pattern. To draw Sierpinski's triangle, start with

a single triangle and connect the midpoints of tlie sides

to draw a smaller triangle. If you repeat this pattern over

and over, you will form a figure like the one shown. This

particular figure started with an isosceles triangle. Are the

triangles outlined in red congruent? Explain.

23. Constructions Use a straightedge to draw any triangleJKL. Then construct AMNP = AJKL using the

given postulate.

a. SSS

b. SAS

Can you prove the triangles congruent? If so, write the congruence statement

and name the postulate you would use. If not, write not enough information and

tell what other information you would need.

24. 25.

N

W

26.

27. Reasoning Suppose GH = }K, HI = KL, and LI = LL. Is AGHI congruent toAJKLI Explain.

M. Given: GK bisects LJGM, GJ = GM

Prove: AGJK= AGMK

M

29. Given: AE and BD bisect each other.

Prove: AACB = AECD

M. Given: FG || KL, FG = KLProof Prove: AFGK s AKLF

31. Given: AB 1 CM, AB 1 DB, CM = DB,Proof M is the midpoint of AB

Prove: AJiMC = AMBD

D

232 Chapter 4 Congruent Triangles

Proof

A Challenge 32. Given: HK = LG, HP = LJ, FG = }KProve: AFGH = AJKL

F

G

33. Given: AN = AL, MN = OL, NO s LMPtoof,

Prove: MN OL

M

N

0

34. Reasoning Four sides of polygon ABCD are congruent, respectively, to the foursides of polygon EFGH. Are ABCD and EFGH congruent? Is a quadrilateral a rigidfigure? If not, what could you add to make it a rigid figure? Explain.

rStandardized Test Prep

35. What additional information do you need to prove that

AVWY = AVWZ by SAS?

CA>W=^ C^AY^AZ

ce::> AWVY = Awvz c^^^vy

36. The measures of two angles of a triangle are 43 and 38. What is themeasure of the third angle?

XSl Ch:>99 CD 100

Short

. Response^iir

37. Which method would you use to find the inverse of a conditional statement?

CD Switch the hypothesis and conclusion. Cp Negate the conclusion only.

CD Negate the hypothesis only. CD Negate both the hypothesis and conclusion.

38. A segment has a midpoint at (1,1) and an endpoint at (—3, 4). What are the

coordinates of tlie other endpoint of the segment? Show your work.

rMixed Review

ABCD = EFGH. Name the angle or side that corresponds to each part.

39. AA 40. W 41. BC

4^ See Lesson 4-1.

42. AG

Write the converse of each statement. Determine whether the statement and its

converse are true or false.

43. If X = 3, then 2x = 6. 44. If x ̂ 3, then = 9.

Get Ready! To prepare for Lesson 4-3^ do Exercises 45 and 46.

45. In AJHK, name the side that is included between A} and AH.

46. In ANLM, name the angle that is included between NM and LN.

^ See Lesson 2-2.

See Lesson 4-2.

c PowerGebmettv.com Lesson 4-2 Triangle Congruence by SSS and SAS 233